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Exact three spin correlation function relations for the square and the honeycomb Ising lattices

In this work, the order parameter and the two-site correlation functions are expressed properly using the decimation transformation process in the presence of an external field so that their applications lead to some significant physical results. Indeed, their applications produce or reproduce some relevant and important results which were included in cumbersome mathematics in the previous studies, if not in a form impossible to understand. The average magnetization or the order parameter $<\!\!σ\!\!> $ is expressed as $<\!\!σ_{0,i}\!\!>= <\!\!\tanh[ κ(σ_{1,i}+σ_{2,i}+\dots +σ_{z,i})+H]\!\!> $. Here, $κ$ is the coupling strength, $z$ is the number of nearest neighbors. $σ_{0,i}$ denotes the central spin at the $i^{th}$ site, while $σ_{l,i}$, $l=1,2,\dots,z$ are the nearest neighbor spins around the central spin. $H$ is the normalized external magnetic field. We show that the application of this relation to the 1D Ising model reproduces readily the previously obtained exact results in the absence of an external field. Furthermore, the three-site correlation functions of square and honeycomb lattices of the form $<\!\!σ_{1}σ_{2}σ_{3}\!\!>$ are analytically obtained. One finds that the three-site correlation functions are equal to $f(κ)\!\!<\!\!σ\!\!>$. Here $f(κ)$ depends on the lattice types and is an analytic function of coupling constant. This result indicates that the critical properties of three-site correlation functions of those lattices are the same as the corresponding order parameters $<\!\!σ\!\!>$ of those lattices. This will mean that the uniqueness of the average magnetization as an order parameter is questionable. ...